#238066

order	512
nilpotency class	3
isoclinism	isoclinic class of #238065
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#393
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

927 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 927 conjugacy classes of subgroups — too many to draw. Only its 170 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,6)(4,7)(5,8)(9,14)(10,13)(11,15)(12,16)(17,18)(19,24)(20,33,36,44)(21,27)(22,26)(23,28)(25,37,42,35)(29,41,46,48)(30,43)(31,39)(32,40)(34,38,45,47);(1,3)(2,6)(4,9)(5,10)(7,14)(8,13)(11,16)(12,15)(17,19)(18,24)(20,34,36,45)(21,31)(22,30)(23,32)(25,41,42,48)(26,43)(27,39)(28,40)(29,35,46,37)(33,47,44,38);(1,4)(2,7)(3,9)(5,11)(6,12)(8,15)(10,16)(13,14)(17,20)(18,25)(19,29)(21,35)(22,37)(23,36)(24,38)(26,33)(27,44)(28,42)(30,45)(31,34)(32,46)(39,48)(40,47)(41,43);(4,11)(7,15)(9,16)(12,14)(17,21,23,22)(18,26,28,27)(19,30,32,31)(20,35,36,37)(24,39,40,43)(25,33,42,44)(29,45,46,34)(38,48,47,41)

Polycyclic presentation:

2^9|a^2=e,b^2=e,d^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,g^a=ghi,c^b=cg,d^b=dh,f^b=fi,d^c=di,e^c=eh,f^c=fh,g^c=gh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 145262);